8.1 Units and the structure of the unit group

Units are the superheroes of number fields, swooping in to save the day with their multiplicative inverses. They form a group that's key to understanding the structure of rings of integers and solving tricky Diophantine equations.

Dirichlet's Unit Theorem is the ultimate cheat code for units. It tells us exactly how many fundamental units we need to generate the whole group, based on the field's embeddings. This powerful tool unlocks deeper insights into number fields.

Units in Number Fields

Defining Units and Their Properties

• Units in a number field K are elements of the ring of integers $O_K$ with multiplicative inverses within $O_K$
• Form a multiplicative group called the unit group of K
• Norm of a unit always equals $\pm1$ due to multiplicative property of norms
• Algebraic integers whose minimal polynomial has a constant term of $\pm1$
• Set of units forms a finitely generated abelian group (Dirichlet's unit theorem)
• Play crucial role in unique factorization of ideals in $O_K$
• Torsion subgroup consists of all roots of unity in the number field

Importance and Applications of Units

• Enable factorization of ideals in $O_K$
• Help determine class number of number fields
• Used in solving Diophantine equations
• Crucial in studying arithmetic properties of number fields
• Aid in constructing integral bases for number fields
• Provide insights into the structure of rings of integers
• Applications in cryptography (elliptic curve cryptosystems)

Structure of Unit Groups

Dirichlet's Unit Theorem

• Unit group isomorphic to direct product of finite cyclic group and free abelian group
• Rank of free abelian part equals $r + s - 1$
  • r: number of real embeddings
  • s: number of pairs of complex embeddings
• Finite cyclic part consists of roots of unity in the number field
• Fundamental units generate entire unit group with roots of unity
• Regulator measures "size" of fundamental units
  • Important in class number formulas
• Real quadratic fields have unit group of rank 1
• Imaginary quadratic fields have finite unit group (only roots of unity)

Examples of Unit Group Structures

• $\mathbb{Q}(\sqrt{2})$: Unit group ${\pm1} \times \langle 1+\sqrt{2} \rangle$
• $\mathbb{Q}(\sqrt{-1})$: Unit group ${\pm1, \pm i}$
• $\mathbb{Q}(\zeta_3)$: Unit group ${\pm1, \pm \zeta_3, \pm \zeta_3^2}$
• Cyclotomic fields $\mathbb{Q}(\zeta_n)$: Unit group generated by cyclotomic units
• Totally real cubic fields: Unit group of rank 2
• CM fields: Unit group structure related to that of its maximal real subfield

Finite Generation of Unit Groups

Logarithmic Embedding Approach

• Proof relies on logarithmic embedding of unit group into real vector space
• Define logarithmic map from unit group to $\mathbb{R}^{r+s}$
  • Maps unit to logarithms of absolute values of its embeddings
• Image contained in hyperplane of dimension $r+s-1$ (product formula)
• Kernel of map is finite (roots of unity in the number field)
• Dirichlet's approximation theorem shows image forms lattice in hyperplane
• Conclude finite generation by combining finite kernel and discrete lattice image

Key Steps in the Proof

• Construct logarithmic map $L: U \to \mathbb{R}^{r+s}$
• Show $\text{Im}(L)$ lies in hyperplane $H: x_1 + \cdots + x_r + 2x_{r+1} + \cdots + 2x_{r+s} = 0$
• Prove $\ker(L)$ is finite using properties of algebraic numbers
• Apply Dirichlet's approximation theorem to find units close to any point in $H$
• Use geometry of numbers to show $\text{Im}(L)$ is a lattice in $H$
• Combine results to prove unit group is finitely generated

Unit Groups: Examples

Quadratic Fields

• Real quadratic fields $\mathbb{Q}(\sqrt{d})$, $d > 0$
  • Find fundamental unit using continued fraction expansion of $\sqrt{d}$
  • Example: $\mathbb{Q}(\sqrt{5})$ has fundamental unit $\frac{1+\sqrt{5}}{2}$
• Imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $d < 0$
  • $\mathbb{Q}(\sqrt{-1})$: Unit group ${\pm1, \pm i}$
  • $\mathbb{Q}(\sqrt{-2})$: Unit group ${\pm1}$
  • $\mathbb{Q}(\sqrt{-3})$: Unit group ${\pm1, \pm \frac{1+\sqrt{-3}}{2}, \pm \frac{-1+\sqrt{-3}}{2}}$

Higher Degree Fields

• Cubic fields: Use norm equations and algebraic manipulations
  • Example: $\mathbb{Q}(\sqrt[3]{2})$ has fundamental unit $\sqrt[3]{2} + 1$
• Employ computational methods (LLL algorithm, Pari/GP software)
• Verify units by checking norms and generation of known small-height units
• Cyclotomic fields $\mathbb{Q}(\zeta_n)$: Describe unit group using cyclotomic units
  • Example: $\mathbb{Q}(\zeta_5)$ has unit group generated by $\zeta_5$ and $\frac{\zeta_5-1}{\zeta_5^2-1}$
• Use regulator and class number formula to verify completeness of fundamental units
  • Example: For real cubic field, compute regulator $R$ and verify $R \approx \frac{h\sqrt{|D|}}{2\pi}$ where $h$ is class number and $D$ is discriminant